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How to know what number of local minima and local maxima is impossible for a polynomial function to have?

For example, the graph of the polynomial $p(x)=ax^5+bx^4+cx^3+dx^2+ex+f$ is sketched. a,b,c,d,e,f are real constants and $a\not=0.$ Which of the following is not possible?

A) The graph has two local minima and two local maxima

B) The graph has one local minimum and two local maxima

C) The graph has one local minimum and one local maximum

D) The graph has no local minima or local maxima

I know that A) and D) are possible, but how can I know whether B) or C) is possible?

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    $\begingroup$ The same question was asked few days back in the same forum. It is duplicate question. $\endgroup$ Aug 19, 2019 at 17:40
  • $\begingroup$ Do you have a link to the question? When I searched for a question similar to mine, I couldn't find anything, so I asked. $\endgroup$ Aug 20, 2019 at 10:07
  • $\begingroup$ I remember some 10days back it was any way solution will be since derivative will be of degree 4 hence Maxima and minimum will occur in pair i.e either one set of local Maxima and minimum , 2 sets of local Maxima and minimum on none of local Maxima and minimum . $\endgroup$ Aug 20, 2019 at 11:03

2 Answers 2

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B) is impossible as the shape of such graph does not matches with the behavior at $\pm$ infinities of a odd degree polynomial.

C) is possible as you can see here

Also this might be useful for you.

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  • $\begingroup$ Thanks! So for an odd function one end is going towards $- $ infinite and another towards $+$ infinite? And for an even function both ends are going towards $+$ or $-$ infinities? $\endgroup$ Aug 19, 2019 at 17:22
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    $\begingroup$ @ErisTyenns: For an even degree polynomial both ends are going towards + infinity or - infinity, depending on the sign of the leading coefficient. $\endgroup$
    – Bumblebee
    Aug 19, 2019 at 17:24
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Hint If a graph has two local maxima and one local minimum, then the minimum must occur between the two minima. On the other hand, if, say, the leftmost extremum of a (nonconstant) polynomial $p$ is a maximum, then as $x \to -\infty$ we also have $p(x) \to -\infty$. With this is mind, what do you know about the behavior of a polynomial of odd degree as $x \to \pm \infty$?

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